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Books > Science & Mathematics > Mathematics > Mathematical foundations > Mathematical logic
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Models of Peano Arithmetic (Hardcover)
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Models of Peano Arithmetic (Hardcover)
Series: Oxford Logic Guides, 15
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Nonstandard models of arithmetic are of interest to mathematicians
through the presence of infinite (or nonstandard) integers and the
various properties they inherit from the finite integers. Since
their introduction in the 1930s (by Skolem and Goedel ), they have
come to play an important role in model theory, and in
combinatorics through independence results such as the
Paris-Harrington theorem. This book is an introduction to these
developments, and stresses the interplay between the first-order
theory, recursion-theoretic aspects, and the structural properties
of these models. Prerequisites have been kept to a minimum. A basic
grounding in elementary model theory and a familiarity with the
notions of recursive, primitive recursive, and r.e. sets will be
sufficient. Consequently, the book should be suitable for
postgraduate students coming to the subject for the first time and
a variety of exercises of varying degrees of difficulty will help
to further the reader's understanding. Beginning with Goedel's
incompleteness theorem, the book covers the prime models, cofinal
extensions, end extensions, Gaifman's construction of a definable
type, Tennenbaum's theorem, Friedman's theorem and subsequent work
on indicators, and culminates in a chapter on recursive saturation
and resplendency.
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